Simulation of the abstract Tile Assembly Model using crisscross slats (extended version)

Tile-based self-assembly is a model of “algorithmic crystal growth” in which square “tiles” represent molecules that bind to each other via specific and variable-strength bonds on their four sides, driven by random mixing in solution but constrained by the local binding rules of the tile bonds. In the late 1990s, Erik Winfree introduced a discrete mathematical model of DNA tile assembly called the abstract Tile Assembly Mode. Winfree proved that the Tile Assembly Model is computationally universal, i.e., that any Turing machine can be encoded into a finite set of tile types whose self-assembly simulates that Turing machine. In this thesis, we investigate tile-based self-assembly systems that exhibit Turing universality, geometric universality and intrinsic universality. We first establish a novel characterization of the computably enumerable languages in terms of self-assembly—the proof of which is a novel proof of the Turing-universality of the Tile Assembly Model in which a particular Turing machine is simulated on all inputs in parallel in the two-dimensional discrete Euclidean plane. Then we prove that the multiple temperature tile assembly model (introduced by Aggarwal, Cheng, Goldwasser, Kao, and Schweller) exhibits a kind of “geometric universality” in the sense that there is a small (constant-size) universal tile set that can be programmed via deliberate changes in the system temperature to uniquely produce any finite shape. Just as other models of computation such as the Turing machine and cellular automaton are known to be “intrinsically universal,” (e.g., Turing machines can simulate other Turing machines, and cellular automata other cellular automata), we show that tile assembly systems satisfying a natural condition known as local consistency are able to simulate other locally consistent tile assembly systems. In other words, we exhibit a particular locally consistent tile assembly system that can simulate the behavior—as opposed to only the final result—of any other locally consistent tile assembly system. Finally, we consider the notion of universal fault-tolerance in algorithmic self-assembly with respect to the two-handed Tile Assembly Model, in which large aggregations of tiles may attach to each other, in contrast to the seeded Tile Assembly Model, in which tiles aggregate one at a time to a single specially-designated “seed” assembly. We introduce a new model of fault-tolerance in self-assembly: the fuzzy temperature model of faults having the following informal characterization: the system temperature is normally 2, but may drift down to 1, allowing unintended temperature-1 growth for an arbitrary period of time. Our main construction, which is a tile set capable of uniquely producing an n × n square with O(log n) unique tile types in the fuzzy temperature model, is not universal but presents novel technique that we hope will ultimately pave the way for a “universal fuzzy-fault-tolerant” tile assembly system in the future.

As a mathematical model of tile-based self-assembling systems, Winfree's abstract Tile Assembly Model (aTAM) has proven to be a remarkable platform for studying and understanding the behaviors and powers of self-assembling systems. Furthermore, as it is capable of Turing computation, the aTAM allows algorithmic self-assembly, in which the components can be designed so that the rules governing their behaviors force them to inherently execute prescribed algorithms as they combine. This power has yielded a wide variety of theoretical results in the aTAM utilizing algorithmic self-assembly to design systems capable of performing complex computations and forming extremely intricate structures. Adding to the completeness of the model, in FOCS 2012 the aTAM was shown to also be intrinsically universal, which means that there exists one single tile set such that for any arbitrary input aTAM system, that tile set can be configured into a seed structure which will then cause self-assembly using that tile set to simulate the input system, capturing its full dynamics modulo only a scale factor. However, the universal of that result makes use of nondeterminism in terms of the tiles placed in several key locations when different assembly sequences are followed. This nondeterminism remains even when the simulator is simulating a system which is directed, meaning that it has exactly one unique terminal assembly and for any given location, no matter which assembly sequence is followed, the same tile type is always placed there. The question which then arose was whether or not that nondeterminism is fundamentally required, and if any simulator must in fact utilize more nondeterminism than directed systems when simulating them. In this paper, we answer that question in the affirmative: the class of directed systems in the aTAM is not intrinsically universal, meaning there is no simulator for directed systems which itself is always directed. This result provides a powerful insight into the role of nondeterminism in self-assembly, which is itself a fundamentally nondeterministic process occurring via unguided local interactions. Furthermore, to achieve this result we leverage powerful results of computational complexity hierarchies, including tight bounds on both best and worst-case complexities of decidable languages, to tailor design systems with precisely controllable space resources available to computations embedded within them. We also develop novel techniques for designing systems containing subsystems with disjoint, mutually exclusive computational powers. The main result will be important in the development of future simulation systems, and the supporting design techniques and lemmas will provide powerful tools for the development of future aTAM systems as well as proofs of their computational abilities.

Self-assembly is a process in which basic units aggregate under attractive forces to form larger compound structures. Recent theoretical work has shown that pseudo-crystalline self-assembly can be algorithmic, in the sense that complex logic can be programmed into the growth process [26]. This theoretical work builds on the theory of two-dimensional tilings [8], using rigid square tiles called Wang tiles [24] for the basic units of self-assembly, and leads to Turing-universal models such as the Tile Assembly Model [28]. Using the Tile Assembly Model, we show how algorithmic self-assembly can be exploited for fabrication tasks such as constructing the patterns that define certain digital circuits, including demultiplexers, RAM arrays, pseudowavelet transforms, and Hadamard transforms. Since DNA self-assembly appears to be promising for implementing the arbitrary Wang tiles [30,13] needed for programming in the Tile Assembly Model, algorithmic self-assembly methods such as those presented in this paper may eventually become a viable method of arranging molecular electronic components [18], such as carbon nanotubes [10,1], into molecular-scale circuits.

Self-assembly is the process whereby relatively simple components autonomously combine to form more complex objects. Nature exhibits self-assembly to form everything from microscopic crystals to living cells to galaxies. With a desire to both form increasingly sophisticated products and to understand the basic components of living systems, scientists have developed and studied artificial self-assembling systems. One such framework is the Tile Assembly Model introduced by Erik Winfree in 1998. In this model, simple two-dimensional square ‘tiles’ are designed so that they self-assemble into desired shapes. The work in this thesis consists of a series of results which build toward the future goal of designing an abstracted, highlevel programming language for designing the molecular components of self-assembling systems which can perform powerful computations and form into intricate structures. The first two sets of results demonstrate self-assembling systems which perform infinite series of computations that characterize computably enumerable and decidable languages, and exhibit tools for algorithmically generating the necessary sets of tiles. In the next chapter, methods for generating tile sets which self-assemble into complicated shapes, namely a class of discrete self-similar fractal structures, are presented. Next, a software package for graphically designing tile sets, simulating their self-assembly, and debugging designed systems is discussed. Finally, a high-level programming language which abstracts much of the complexity and tedium of designing such systems, while preventing many of the common errors, is presented. The summation of this body of work presents a broad coverage of the spectrum of desired outputs from artificial self-assembling systems and a progression in the sophistication of tools used to design them. By creating a broader and deeper set of modular tools for designing xii self-assembling systems, we hope to increase the complexity which is attainable. These tools provide a solid foundation for future work in both the Tile Assembly Model and explorations into more advanced models.

We prove a negative result on the power of a model of algorithmic self-assembly for which finding general techniques and results has been notoriously difficult. Specifically, we prove that Winfree's abstract Tile Assembly Model is not intrinsically universal when restricted to use noncooperative tile binding. This stands in stark contrast to the recent result that the abstract Tile Assembly Model is indeed intrinsically universal when cooperative binding is used (FOCS 2012). Noncooperative self-assembly, also known as “temperature 1”, is where all tiles bind to each other if they match on at least one side. On the other hand, cooperative self-assembly requires that some tiles bind on at least two sides. Our result shows that the change from non-cooperative to cooperative binding qualitatively improves the range of dynamics and behaviors found in these models of nanoscale self-assembly. The result holds in both two and three dimensions; the latter being quite surprising given that three-dimensional noncooperative tile assembly systems simulate Turing machines. This shows that Turing universal behavior in self-assembly does not imply the ability to simulate all algorithmic self-assembly processes. In addition to the negative result, we exhibit a three-dimensional noncooperative self-assembly tile set capable of simulating any two-dimensional noncooperative self-assembly system. This tile set implies that, in a restricted sense, non-cooperative self-assembly is intrinsically universal for itself.

We prove a negative result on the power of a model of algorithmic self-assembly for which finding general techniques and results has been notoriously difficult. Specifically, we prove that Winfree's abstract Tile Assembly Model is not intrinsically universal when restricted to use noncooperative tile binding. This stands in stark contrast to the recent result that the abstract Tile Assembly Model is indeed intrinsically universal when cooperative binding is used (FOCS 2012). Noncooperative self-assembly, also known as temperature 1, is where all tiles bind to each other if they match on at least one side. On the other hand, cooperative self-assembly requires that some tiles bind on at least two sides.Our result shows that the change from non-cooperative to cooperative binding qualitatively improves the range of dynamics and behaviors found in these models of nanoscale self-assembly. The result holds in both two and three dimensions; the latter being quite surprising given that three-dimensional noncooperative tile assembly systems simulate Turing machines. This shows that Turing universal behavior in self-assembly does not imply the ability to simulate all algorithmic self-assembly processes. In addition to the negative result, we exhibit a three-dimensional noncooperative self-assembly tile set capable of simulating any two-dimensional noncooperative self-assembly system. This tile set implies that, in a restricted sense, non-cooperative self-assembly is intrinsically universal for itself.

In this paper we present a series of results related to mathematical models of self-assembling systems of tiles and the impacts that three diverse properties have on their dynamics. In these self-assembling systems, initially unorganized collections of tiles undergo random motion and can bind together, if they collide and enough of their incident glues match, to form assemblies. Here we greatly expand upon a series of prior results which showed that (1) the abstract Tile Assembly Model (aTAM) is intrinsically universal (FOCS 2012), and (2) the class of directed aTAM systems is not intrinsically universal (FOCS 2016). Intrinsic universality (IU) for a model (or class of systems within a model) means that there is a universal tile set which can be used to simulate an arbitrary system within that model (or class). Furthermore, the simulation must not only produce the same resultant structures, it must also maintain the full dynamics of the systems being simulated and display the same behaviors modulo a scale factor. While the FOCS 2012 result showed that the standard, two-dimensional (2D) aTAM is IU, here we show that this is also the case for the three-dimensional (3D) version. Conversely, the FOCS 2016 result showed that the class of aTAM systems which are directed (a.k.a. deterministic, or confluent) is not IU, meaning that there is no universal simulator which can simulate directed aTAM systems while itself always remaining directed, implying that nondeterminism is fundamentally required for such simulations. Here, however, we show that, in 3D, the class of directed aTAM systems is actually IU, i.e. there is a universal directed simulator for them. This implies that the constraint of tiles binding only in the plane forced the necessity of nondeterminism for the simulation of 2D directed systems. This then leads us to continue to explore the impacts of dimensionality and directedness on simulation of tile-based self-assembling systems by considering the influence of more rigid notions of dimensionality. Namely, we introduce the Planar aTAM, where tiles are not only restricted to binding in the plane, but they are also restricted to traveling within the plane, and we prove that the Planar aTAM is not IU, and prove that the class of directed systems within the Planar aTAM also is not IU. Finally, analogous to the Planar aTAM, we introduce the Spatial aTAM, its 3D counterpart, and prove that the Spatial aTAM is IU. This paper adds to a broad set of results which have been used to classify and compare the relative powers of differing models and classes of self-assembling systems, and also helps to further the understanding of the roles of dimension and nondeterminism on the dynamics of self-assembling systems. Furthermore, to prove our positive results we have not only designed, but also implemented what we believe to be the first IU tile set ever implemented and simulated in any tile assembly model, and have made it, along with a simulator which can demonstrate it, freely available.

Self-assembly is fundamental to both biological processes and nanoscience. Key features of self-assembly are its probabilistic nature and local programmability. These features can be leveraged to design better self-assembled systems. The conventional tile assembly model (TAM) developed by Winfree using Wang tiles is a powerful, Turing-universal theoretical framework which models varied self-assembly processes. A particular challenge in DNA nanoscience is to form linear assemblies or rulers of a specified length using the smallest possible tile set, where any tile type may appear more than once in the assembly. The tile complexity of a linear assembly is the cardinality of the tile set that produces it. These rulers can then be used as components for construction of other complex structures. While square assemblies have been extensively studied, many questions remain about fixed length linear assemblies, which are more basic constructs yet fundamental building blocks for molecular architectures. In this work, we extend TAM to take advantage of inherent probabilistic behavior in physically realized self-assembled systems by introducing randomization. We describe a natural extension to TAM called the probabilistic tile assembly model (PTAM). A restriction of the model, which we call the standard PTAM is considered in this report. Prior work in DNA self-assembly strongly suggests that standard PTAM can be realized in the laboratory. In TAM, a deterministic linear assembly of length $N$ requires a tile set of cardinality at least $N$. In contrast, we show various nontrivial probabilistic constructions for forming linear assemblies in PTAM with tile sets of sublinear cardinality, using techniques that differ considerably from existing assembly techniques. In particular, for any given $N$ we demonstrate linear assemblies of expected length $N$ with a tile set of cardinality $\Theta(\log N)$ using one pad per side of each tile. We prove a matching lower bound of $\Omega(\log N)$ on the tile complexity of linear assemblies of any given expected length $N$ in standard PTAM systems using one pad per side of each tile. We further demonstrate how linear assemblies can be modified to produce assemblies with sharp tail bounds on distribution of lengths by concatenating various assemblies together. In particular, we show that for infinitely many $N$ we can get linear assemblies with exponentially dropping tail distributions using $O(\log^3 N)$ tile types. We also propose a simple extension to PTAM called $\kappa$-pad systems in which we associate $\kappa$ pads with each side of a tile, allowing abutting tiles to bind when at least one pair of corresponding pads match. This gives linear assemblies of expected length $N$ with a $2$-pad (two pads per side of each tile) tile set of cardinality $\Theta(\frac{\log N}{\log \log N})$ for infinitely many $N$. We show that we cannot get smaller tile complexity by proving a lower bound of $\Omega(\frac{\log N}{\log \log N})$ for each $N$ on the cardinality of the $\kappa$-pad ($\kappa$-pads per side of each tile) tile set required to form linear assemblies of expected length $N$ in standard $\kappa$-pad PTAM systems for any positive integer $\kappa$. The techniques that we use for deriving these tile complexity lower bounds are notable as they differ from traditional Kolmogorov complexity based information theoretic methods used for lower bounds on tile complexity. Also, Kolmogorov complexity based lower bounds do not preclude the possibility of achieving assemblies of very small tile multiset cardinality for infinitely many $N$. In contrast, our lower bounds are stronger as they hold for every $N$, rather than for almost all $N$. All our probabilistic constructions are free from co-operative tile binding errors. Thus, for linear assembly systems, we have shown that randomization can be exploited to get large improvements in tile complexity at a small expense of precision in length.

Self-assembly is a powerful process found in nature that guides simple objects assembling, on their own, into complex structures. Self-assembly is of interest to computer scientists because self-assembling systems can compute functions, assemble shapes, and guide distributed robotics systems. The tile assembly model is a formal mathematical model of self-assembly that allows the study of time and space complexities of self-assembling systems that lie at the heart of several molecular computer implementations and distributed computational software systems. These implementations and systems require efficient tile systems with small tilesets and fast execution times. The state of the art, however, requires vastly complex tile systems with large tilesets to implement fast algorithms. In this paper, I present \({\mathbb{S}}_{FS},\) a tile system that decides 3-SAT by creating \(O^{\star}(1.8393^n)\) nondeterministic assemblies in parallel, improving on the previous best known solution that requires \(\Uptheta(2^n)\) such assemblies. This solution directly improves the feasibility of building molecular 3-SAT solvers and efficiency of distributed software. I formally prove the correctness of the system, the number of required parallel assemblies, that the size of the system’s tileset is \(147 = \Uptheta(1),\) and that the assembly time is nondeterministic linear in the size of the input.

Self-assembly is the ubiquitous process by which objects autonomously assemble into complexes. This phenomenon is common in nature and yet is poorly understood from mathematical and programming perspectives. It is believed that self-assembly technology will ultimately permit the precise fabrication of complex nanostructures. Of particular interest is DNA self-assembly. Double and triple crossover DNA molecules have been designed that can act as four-sided building blocks for DNA self-assembly. Experimental work has been done to show the effectiveness of using these building blocks to assemble DNA crystals and perform DNA computation. With these building blocks (called tiles) in mind, researchers have considered the power of the tile self-assembly model. The tile assembly model extends the theory of Wang tilings of the plane by adding a natural mechanism for growth. Informally, the model consists of a set of four sided Wang tiles whose sides are each associated with a type of glue. The bonding strength between any two glues is determined by a glue function. A special tile in the tile set is denoted as the seed tile. Assembly takes place by starting with the seed tile and attaching copies of tiles from the tile set one by one to the growing seed whenever the total strength of attraction from the glue function meets or exceeds a fixed parameter called the temperature. Algorithmic DNA self-assembly is both a form of nanotechnology and a model of DNA computing. As a computational model, algorithmic DNA self-assembly encodes the input of a computational problem into DNA patterns and then manipulates these patterns to produce new DNA patterns that encode the desired output of the computational problem. As a nanotechnology, algorithmic DNA self-assembly aims to design tiles with carefully chosen glue types on their four sides. Two tiles are said to be of different types if their sides have different glue types. Useful tile types are nontrivial to design but relatively easy to duplicate in large quantity. A key design challenge for algorithmic DNA self-assembly is to use only a small number of different tile types to assemble a target nanostructure. This talk will survey recent results in algorithmic DNA self-assembly and discuss future research directions.

The field of algorithmic self-assembly is concerned with the computational and expressive power of nanoscale self-assembling molecular systems. In the well-studied cooperative, or temperature 2, abstract tile assembly model it is known that there is a tile set to simulate any Turing machine and an intrinsically universal tile set that simulates the shapes and dynamics of any instance of the model, up to spatial rescaling. It has been an open question as to whether the seemingly simpler noncooperative, or temperature 1, model is capable of such behaviour. Here we show that this is not the case by showing that there is no tile set in the noncooperative model that is intrinsically universal, nor one capable of time-bounded Turing machine simulation within a bounded region of the plane. Although the noncooperative model intuitively seems to lack the complexity and power of the cooperative model it has been exceedingly hard to prove this. One reason is that there have been few tools to analyse the structure of complicated paths in the plane. This paper provides a number of such tools. A second reason is that almost every obvious and small generalisation to the model (e.g. allowing error, 3D, non-square tiles, signals/wires on tiles, tiles that repel each other, parallel synchronous growth) endows it with great computational, and sometimes simulation, power. Our main results show that all of these generalisations provably increase computational and/or simulation power. Our results hold for both deterministic and nondeterministic noncooperative systems. Our first main result stands in stark contrast with the fact that for both the cooperative tile assembly model, and for 3D noncooperative tile assembly, there are respective intrinsically universal tilesets. Our second main result gives a new technique (reduction to simulation) for proving negative results about computation in tile assembly.

The tile assembly model has allowed the study of the nature's process of self-assembly and the development of self-assembling systems for solving complex computational problems. Research into this model has led to progress in two distinct classes of computational systems: Internet-sized distributed computation, such as software architectures for computational grids, and molecular computation, such as DNA computing. The design of large complex tile systems that emulate Turing machines has shown that the tile assembly model is Turing universal, while the design of small tile systems that implement simple algorithms has shown that tile assembly can be used to build private, fault-tolerant, and scalable distributed software systems and robust molecular machines. However, in order for these types of systems to compete with traditional computing devices, we must demonstrate that fairly simple tile systems can implement complex and intricate algorithms for important problems. The state of the art, however, requires vastly complex tile systems with large tile sets to implement such algorithms. In this paper, I present SFS, a tile system that decides 3-SAT by creating O*(1.8393n) nondeterministic assemblies in parallel, while the previous best known solution requires Θ(2n) such assemblies. In some sense, this tile system follows the most complex algorithm implemented using tiles to date. I analyze that the number of required parallel assemblies is O*(1.8393n), that the size of the system's tileset is 147 = Θ(1), and that the assembly time is nondeterministic linear in the size of the input. This work directly improves the time and space complexities of tile-inspired computational-grid architectures and bridges theory and today's experimental limitations of DNA computing.

The paper considers molecular programming in the abstract Tile Assembly Model, aTAM. Using simple constructions, an interpreter for the full Combinatory Logic, CL, is formally defined in aTAM. It provides an approach for sequential programming in aTAM, and produces a DNA molecular machine. This machine lives in a suitable solution and when receives a seed that linearly encodes a CL program and an input for the program, produces a grid which encodes a computation of the program on its input. The paper considers the construction cost and some alternative approaches.Finally, as a case study in distributed programming in aTAM, the paper considers the consensus problem and shows how an aTAM program for it can be formally derived by using Pi-calculus

In this paper we consider the time complexity of adding two n-bit numbers together within the tile self-assembly model. The (abstract) tile assembly model is a mathematical model of self-assembly in which system components are square tiles with different glue types assigned to tile edges. Assembly is driven by the attachment of singleton tiles to a growing seed assembly when the net force of glue attraction for a tile exceeds some fixed threshold. Within this frame work, we examine the time complexity of computing the sum of two n-bit numbers, where the input numbers are encoded in an initial seed assembly, and the output sum is encoded in the final, terminal assembly of the system. We show that this problem, along with multiplication, has a worst case lower bound of $$\varOmega ( \sqrt{n} )$$Ω(n) in 2D assembly, and $$\varOmega (\root 3 \of {n})$$Ω(n3) in 3D assembly. We further design algorithms for both 2D and 3D that meet this bound with worst case run times of $$O(\sqrt{n})$$O(n) and $$O(\root 3 \of {n})$$O(n3) respectively, which beats the previous best known upper bound of O(n). Finally, we consider average case complexity of addition over uniformly distributed n-bit strings and show how we can achieve $$O(\log n)$$O(logn) average case time with a simultaneous $$O(\sqrt{n})$$O(n) worst case run time in 2D. As additional evidence for the speed of our algorithms, we implement our algorithms, along with the simpler O(n) time algorithm, into a probabilistic run-time simulator and compare the timing results.

In this paper we consider the time complexity of adding two n-bit numbers together within the tile self-assembly model. The (abstract) tile assembly model is a mathematical model of self-assembly in which system components are square tiles with different glue types assigned to tile edges. Assembly is driven by the attachment of singleton tiles to a growing seed assembly when the net force of glue attraction for a tile exceeds some fixed threshold. Within this frame work, we examine the time complexity of computing the sum of 2 n-bit numbers, where the input numbers are encoded in an initial seed assembly, and the output sum is encoded in the final, terminal assembly of the system. We show that this problem, along with multiplication, has a worst case lower bound of \(\Omega( \sqrt{n} )\) in 2D assembly, and \(\Omega(\sqrt[3]{n})\) in 3D assembly. We further design algorithms for both 2D and 3D that meet this bound with worst case run times of \(O(\sqrt{n})\) and \(O(\sqrt[3]{n})\) respectively, which beats the previous best known upper bound of O(n). Finally, we consider average case complexity of addition over uniformly distributed n-bit strings and show how we can achieve O(logn) average case time with a simultaneous \(O(\sqrt{n})\) worst case run time in 2D. As additional evidence for the speed of our algorithms, we implement our algorithms, along with the simpler O(n) time algorithm, into a probabilistic run-time simulator and compare the timing results.KeywordsBond GraphTile Assembly ModelInput TemplateTile Assembly SystemSeed AssemblyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.